
6. Miscellaneous problems

1. Hermite-Hadamard inequality in higher dimensions

The classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then $$\frac{1}{b-a}\int_a^b f(x)dx \leq \frac{f(b)+f(a)}{2}.$$ We consider two generalizations to higher dimensions. Let $\Omega \subset \mathbb{R}^n$ be a convex domain and

Case 1: $f\colon \Omega \to [0, \infty)$ convex, or

Case 2: $f\colon \Omega \to [0, \infty)$ subharmonic.

In either case one can ask for smallest constants $c_n, \tilde c_n$ such that $$\frac{1}{|\Omega|}\int_\Omega f(x)dx \leq \frac{c_n}{|\partial\Omega|}\int_{\partial\Omega}f(x)dx, \quad\mbox{and}\quad \int_\Omega f(x)dx \leq \tilde c_n|\Omega|^{1/n}\int_{\partial\Omega}f(x)dx.$$ The second inequality follows from the first one by an application of the isoperimetric inequality with $\tilde c_n = c_n/\sqrt{n}$.

Problem 6.1.

[S. Steinerberger] Find the optimal constants in the inequalities.
Recently, S. Steinerberger [MR4058521] proved that in the case of convex functions $c_n \leq 2 \pi^{-1/2}n^{n+1}$ and that in the two-dimensional case $9/8\leq c_2 \leq 8$. For subharmonic functions J. Lu and S. Steinerberger [MR4052204] showed that $\tilde c_n \leq 1$. In [arXiv:1804.03688] P. Pasteczka conjectured that the constant $c_n$ in the case of convex functions can be taken to be $1$ if the centre of mass of $\Omega$ coincides with that of $\partial\Omega$. That the centres of mass of $\Omega$ and $\partial\Omega$ coincide is known to be a necessary condition for $c_n = 1$ by testing against affine functions.

During the workshop improved bounds were found for the sharp constants in all cases [arXiv:1907.06122].
• Asymptotic partitioning problems

In several problems where one wants to find an in some sense optimal partition of a planar domain $\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.

Problem 6.2.

[K. Burdzy] Find conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\to \infty$ a packing of hexagons.
The phenomenon has been proved by D. Bucur and I. Fragala if the optimality is with respect to minimization for the sum of the Cheeger constants [MR3946306] or minimization for the sum of the lowest Robin eigenvalues [MR3918044] of the partition elements. The original conjecture formulated by L. Caffarelli and F. Lin in [MR2304268] for the optimality with respect to minimization for the sum of the principal Dirichlet eigenvalues is still open.
• The ovals of Benguria and Loss

Let $\Gamma\subset \mathbb{R}^2$ be a closed smooth curve of length $2\pi$ and let $\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator $H_\Gamma \psi := -\psi'' + \kappa^2\psi,\qquad \mathsf{dom}\,H_\Gamma := W^{2,2}(\Gamma),$ in the Hilbert space $L^2(\Gamma)$ and let $\lambda_1(H_\Gamma)$ be its lowest eigenvalue.

Conjecture 6.3.

[R. Benguria] The following inequality $\lambda_1(H_\Gamma) \ge 1$ holds.
The problem has appeared in connection to sharp Lieb-Thirring inequalities for Schrödinger operators with two bound states. R. Benguria and M. Loss proved in [MR2091490] that $\lambda_1(H_\Gamma)\geq 1/2$ which has later been improved by H. Linde in [MR2240676] upto $\lambda_1(H_\Gamma)> 0.6085$. It is also shown in [MR2203162] that there is an infinite family of ovals for which $\lambda_1(H_\Gamma)=1$, which includes the circle and the "double segment". S. Steinerberger remarked that N. Marshall has noticed that the curvature flow defined by $$\frac{d}{dt}\kappa(s, t) = \kappa(s, t)\frac{1}{2\pi}\int_0^{2\pi}\kappa(x, t)^2dx - \kappa(s, t)^2$$ preserves the family of ovals which are suggested as minimizers. However, the flow might break up curves. This flow or its modification can be useful for the proof of the conjecture.
• Faber-Krahn for the buckling problem

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem \begin{align} \Delta^2 u+\Lambda_1(\Omega)\Delta u &=0 \quad \mbox{in }\Omega,\\ u&=0 \quad \mbox{on }\partial\Omega,\\ \frac{\partial u}{\partial n}&=0 \quad \mbox{on }\partial\Omega. \end{align}

Problem 6.4.

[D. Bucur] Prove that the lowest buckling eigenvalue $\Lambda_1(\Omega)$ is minimized by the disk among all simply connected sets of equal area.
The problem is related to clamped plate problem where corresponding result has been proved by N. Nadirashvili [MR1328469] and by M. Ashbaugh and R. Benguria [MR1703569]. Existence of a minimizer for the buckling problem is known [MR2019178].

D. Bucur remarks that the problem can be reformulated in terms of the eigenvalue problem \begin{align} -\Delta u+ \Lambda_1(\Omega)(u-P_\Omega u) &=0 \quad \mbox{in }\Omega,\\ u&=0 \quad \mbox{on }\partial\Omega, \end{align} where $P_\Omega$ is an orthogonal projection from $L^2(\Omega)$ into a subspace of harmonic functions. If the subspace is that of all harmonic functions one retains the eigenvalue of the buckling problem.
1. Remark. This buckling load conjecture was raised by G. Polya and G. Szego in their book [MR0043486] on isoperimetric inequalities in mathematical physics.
• Poincaré-Wirtinger extremal domain

Consider the operator $-\gamma^{-1}\nabla \gamma \nabla$ on the weighted space $L^2(\Omega, \gamma)$ with Neumann boundary conditions, $\Omega\subset \mathbb{R}^n$ and where $\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\mu_1(\Omega, \gamma)\geq 1$ (this is the Poincaré-Wirtinger inequality).

Problem 6.5.

[D. Krejcirik] Prove that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.
With a priori assumption that $\Omega$ is contained in an infinite strip the result is known to be true [MR3556341].
• Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains

Consider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.

Problem 6.6.

[K. Burdzy] Under what assumptions can one prove that the nodal domains are in a sense ordered?
There are related works in the Dirichlet setting by P. Freitas and D. Krejcirik [MR2400260] for tubular neighbourhoods of curves by D. Krejcirik and M. Tusek [MR3274759] for tubular neighbourhoods of hypersurfaces.

Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.